Question

Find the critical points of the following function. Use the Second Derivative Test to determine​ (if possible) whether each critical point corresponds to a local​ maximum, local​ minimum, or saddle point. Confirm your results with a graphing utility.

F(x,y) = x² + 2x²(y-2)+7(y-1)²

Answer 1

To find the critical points of the function F(x,y), we calculate the first derivatives, set them to zero to solve for critical points, then use the Second Derivative Test to determine whether each point represents a local maximum, minimum, or saddle point. Lastly, these findings can be visually confirmed using a graphing utility.

Step-by-step explanation:

Finding Critical Points and Their Nature

To find the critical points of the function F(x,y) = x² + 2x²(y-2)+7(y-1)², we first need to find the first derivatives F_x and F_y and set them equal to zero. F_x = 2x + 4x(y - 2), and F_y = 4x² - 14(y - 1). Setting F_x and F_y to zero gives us the system of equations:

• 2x + 4x(y - 2) = 0
• 4x² - 14(y - 1) = 0

Solving this system, we find the critical points. To determine the nature of the critical points, we apply the Second Derivative Test by finding the second derivatives F_xx, F_yy, and F_xy, and then evaluating the determinant D = F_xx * F_yy - (F_xy)² at the critical points. Depending on the sign of D, and provided that F_xx and F_yy are nonzero, we can determine if a critical point is a local maximum, local minimum, or a saddle point.

Lastly, to confirm the results, we can use a graphing utility to plot the function and visually inspect the behavior around the critical points to ensure it aligns with our calculations.